Generating an output for a neural network output layer

ABSTRACT

Systems, methods, and apparatus, including computer programs encoded on a computer storage medium for processing a network input through a neural network having one or more initial neural network layers followed by a softmax output layer. In one aspect, the methods include obtaining a layer output generated by the one or more initial neural network layers and processing the layer output through the softmax output layer to generate a neural network output. Processing the layer output through the softmax output layer includes determining, for each possible output value, a number of occurrences in the layer output values; for each possible output value occurring in the layer output values, determining a respective exponentiation measure; determining a normalization factor for the layer output by combining the exponentiation measures in accordance with the number of occurrences of the possible output values; and determining, for each of layer output values, a softmax probability value.

BACKGROUND

This specification relates to generating outputs for neural networkoutput layers.

Neural networks are machine learning models that employ one or morelayers of nonlinear units to predict an output for a received input.Some neural networks include one or more hidden layers in addition to anoutput layer. The output of each hidden layer is used as input toanother layer in the network, i.e., the next hidden layer or the outputlayer. Each layer of the network generates an output from a receivedinput in accordance with current values of a respective set ofparameters.

SUMMARY

This specification describes how a system can process the output of aneural network. To do so, the system determines the number ofoccurrences of each member of a finite set of potential output valuesamong the output generated by the initial neural network layers of theneural network. The system determines a softmax layer output for eachvalue occurring in the output of the initial neural network layers bydetermining a respective exponentiation measure for each occurringvalue.

In general, one innovative aspect of the subject matter described inthis specification can be embodied in methods of processing a networkinput through a neural network having one or more initial neural networklayers followed by a softmax output that include the actions ofobtaining a layer output generated by processing the network inputthrough the one or more initial neural network layers, the layer outputhaving a plurality of layer output values and each layer output valuebeing a respective one of a predetermined finite set of possible outputvalues; and processing the layer output through the softmax output layerto generate a neural network output for the network input, includingdetermining, for each possible output value in the predetermined finiteset, a number of occurrences of the possible output value in theplurality of layer output values; for each possible output valueoccurring in the plurality of layer output values, determining arespective exponentiation measure of the possible output value;determining a normalization factor for the layer output by combining theexponentiation measures in accordance with the number of occurrences ofthe possible output values; and determining, for each of the pluralityof layer output values, a softmax probability value from the respectiveexponentiation measure for the layer output value and the normalizationfactor.

Other embodiments of this aspect include corresponding computer systems,apparatus, and computer programs recorded on one or more computerstorage devices, each configured to perform the actions of the methods.A system of one or more computers can be configured to performparticular operations or actions by virtue of software, firmware,hardware, or any combination thereof installed on the system that inoperation may cause the system to perform the actions. One or morecomputer programs can be configured to perform particular operations oractions by virtue of including instructions that, when executed by dataprocessing apparatus, cause the apparatus to perform the actions.

The foregoing and other embodiments can each optionally include one ormore of the following features, alone or in combination. In particular,one embodiment includes all the following features in combination.

In some implementations, obtaining the layer output includes receiving aplurality of initial layer output values from a processing system thatdoes the processing for the one or more initial neural network layers,the plurality of initial layer output values being unmapped outputvalues of the one or more initial neural network layers; obtainingmapping data defining a mapping from the plurality of initial layeroutput values to the plurality of layer output values; and determining,for each initial layer output value, a layer output value based on themapping data. In some of those implementations, the mapping dataspecifies a scaling factor for scaling each of the plurality of initiallayer output values to generate the layer output values.

In some implementations, the layer output is generated by a processingdevice that performs computation specified by the one or more initialneural network layers using quantized arithmetic. In someimplementations, the layer output is generated by a processing devicethat performs computation specified by the one or more initial neuralnetwork layers using fixed-point arithmetic.

In some implementations, each of the finite set of possible outputvalues map to a respective value of an integer data type. In someimplementations, the methods further include generating the networkinput by converting one or more floating point values to fixed pointvalues. In some implementations, determining the respectiveexponentiation measure of the possible output value includesexponentiating Euler's number by a multiplication of each respectivepossible output value. In some implementations, each softmax probabilityvalue is determined by dividing each respective exponentiation measureby the normalization factor.

In some implementations, each of the finite set of possible outputvalues is an output of a mapping function. In some implementations, eachof the finite set of possible output values is an output of acompression function.

Particular embodiments of the subject matter described in thisspecification can be implemented so as to realize one or more of thefollowing advantages. The computational complexity of determining anoutput of a softmax output layer of a neural network can be reduced. Inprocessing systems that perform computations using quantized arithmetic,the range of potential values on which operations can be performed islimited by the range of countable values to which a set of values aremapped. Particular implementations of the subject matter described inthis specification leverage this quality of such processing systems bypre-computing normalized values for countable values that occur amongthe output of the neural network, thus increasing the efficiency ofnormalizing output values of neural networks. In these implementations,precomputing the value of exponentiation measures needed for computingnormalized values can eliminate the need for hardware or softwaresupport for exponentiation operations.

The details of one or more embodiments of the subject matter of thisspecification are set forth in the accompanying drawings and thedescription below. Other features, aspects, and advantages of thesubject matter will become apparent from the description, the drawings,and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an example neural network processing system.

FIG. 2 is a flow chart of an example process for generating a networkoutput from a layer output.

FIG. 3 is a flow chart of an example process for mapping initial layeroutput values to layer output values.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

FIG. 1 shows an example neural network processing system 100. The neuralnetwork processing system 100 is an example of a system implemented ascomputer programs on one or more computers in one or more locations, inwhich the systems, components, and techniques described below areimplemented.

The neural network processing system 100 includes a neural network 101.The neural network 101 receives a network input 141 and processes theinput 141 to generate a network output 161. The neural networkprocessing system 100 can store the generated network output 161 in anoutput data repository or provide the network output for use for someother immediate purpose, e.g., for presentation on a user device or forfurther processing by another system.

The neural network 101 can be configured to receive any kind of digitaldata input and to generate any kind of score or classification outputbased on the input.

For example, if the inputs to the neural network 101 are images orfeatures that have been extracted from images, the output generated bythe neural network 101 for a given image may be scores for each of a setof object categories, with each score representing an estimatedlikelihood that the image contains an image of an object belonging tothe category.

As another example, if the inputs to the neural network 101 are Internetresources (e.g., web pages), documents, or portions of documents orfeatures extracted from Internet resources, documents, or portions ofdocuments, the output generated by the neural network 101 for a givenInternet resource, document, or portion of a document may be a score foreach of a set of topics, with each score representing an estimatedlikelihood that the Internet resource, document, or document portion isabout the topic.

As another example, if the inputs to the neural network processingsystem 100 are features of an impression context for a particularadvertisement, the output generated by the neural network 101 may be ascore that represents an estimated likelihood that the particularadvertisement will be clicked on.

As another example, if the inputs to the neural network 101 are featuresof a personalized recommendation for a user, e.g., featurescharacterizing the context for the recommendation, e.g., featurescharacterizing previous actions taken by the user, the output generatedby the neural network 101 may be a score for each of a set of contentitems, with each score representing an estimated likelihood that theuser will respond favorably to being recommended the content item.

As another example, if the input to the neural network 101 is text inone language, the output generated by the neural network 101 may be ascore for each of a set of pieces of text in another language, with eachscore representing an estimated likelihood that the piece of text in theother language is a proper translation of the input text into the otherlanguage.

As another example, if the input to the neural network 101 is a spokenutterance, a sequence of spoken utterances, or features derived from oneof the two, the output generated by the neural network 101 may be ascore for each of a set of pieces of text, each score representing anestimated likelihood that the piece of text is the correct transcriptfor the utterance or sequence of utterances.

In particular, the neural network 101 includes one or more initialneural network layers 110 and a softmax output layer 120. The initialneural network layers 110 can be the input 151 and hidden 152 and 153 ofa feedforward or recurrent neural network. The initial neural networklayers 110 are configured to receive a neural network input and processthe neural network input to generate an initial layer output 160.Generally, the initial layer output 160 is a vector or other orderedcollection of numeric values that includes a predetermined number oflayer output values, i.e., the number of values that the final initiallayer is configured to output.

The softmax output layer 120 is configured to receive the initial layeroutput 160 and generate a network output 161 based on the initial layeroutput 160 by applying a softmax function to the layer output 160.

In some implementations, the neural network processing system 100modifies the initial layer output 160 prior to the initial layer output160 being processed by the softmax output layer 120. In particular, inthese implementations, the neural network processing system 100 modifiesthe layer output 160 by mapping each initial layer output value toanother output value using mapping data. The mapping data may map eachvalue in the initial layer output 160 to a different value, e.g., a morecomplex value, or include a scaling factor to be applied to the valuesin the initial layer output 160 before the values are processed by thesoftmax output layer 120. Mapping initial layer output values before thevalues are processed by a softmax output layer is described in greaterdetail below with reference to FIG. 3.

Generally, the softmax function normalizes the layer output 160 so eachvalue in the network output 161 is a value within a predefined range(e.g., the predefined range of real values between 0 and 1, inclusive).The softmax function can also be referred to as the normalizedexponential function.

In particular, the neural network processing system 100 processes thenetwork input 141 through the initial neural network layers 110 suchthat each value in the layer output 160 or, in implementations in whichthe neural network processing system 100 modifies the values in thelayer output 160, each modified layer output value belongs to apredetermined finite set of possible values. Reasons for why the layeroutput 160 may have this characteristic are described below withreference to FIG. 2.

Because of this, the neural network processing system 100 can processthe layer output 160 through the softmax output layer 120 in anefficient manner by computing a count of each of the possible values inthe predetermined range and calculating exponentiation measures andnormalization factors for the occurring values only once. Generating acorresponding network output 161 for a given layer output 160 in thismanner is described in more detail below with reference to FIG. 2.

FIG. 2 is a flow chart of an example process 200 for generating anetwork output from a layer output. For convenience, the process 200will be described as being performed by a system of one or morecomputers located in one or more locations. For example, a neuralnetwork processing system, e.g., the neural network processing system100 of FIG. 1, appropriately programmed in accordance with thisspecification, can perform the process 200.

The system obtains a layer output generated by processing a networkinput through the one or more initial neural network layers (210). Thelayer output includes a predetermined number of layer output values,where each layer output value is a respective one of a predeterminedfinite set of possible output values. The predetermined finite set ofpossible output values is a finite set of values that may include asmaller number of values than the number of values that may be suppliedto a softmax function over a period of time.

The layer output values may include only values from the finite set as aresult of the manner in which the processing device generating the layeroutput performs computations. In some implementations, the layer outputis generated by a processing device that performs computation specifiedby the one or more initial neural network layers with reduced precision,i.e., using quantized arithmetic.

Quantized arithmetic involves representing a first set of values by asecond set of values, where the second set includes fewer values thanthe first set. For instance, the first set of values may be values of areal data type while the second set may be values of a fixed point datatype. Fixed point data types represent real values by a number that hasa fixed number of digits after the radix point and as such can limit thesize of each stored value and the range of possible values that can bestored. In processing systems that use quantized arithmetic, the secondset of values to which the values of the first set are mapped mayinclude a predetermined finite set of possible values.

In some implementations, the finite set of possible output valuesinclude the set of values of an integer data type that has a boundednumber of bits. Examples of the predetermined finite set of possibleoutput values include the integers included in the interval [−128, 127],if the system generates layer outputs that are of an 8-bit signedinteger type, or the integers included in the interval [0, 255], if thesystem generates layer outputs that are of an 8-bit unsigned integertype. The predetermined finite set of output values may include valuesof any one or more data types. In exemplary implementations, thepredetermined finite set of possible output values may include integersin the interval [0, 100], all ASCII characters, and the following realnumbers: 1.1, 2.2, 3.3, 4.4, 5.5, 6.6, 7.7, 8.8, and 9.9.

The system determines, for each possible output value in thepredetermined finite set, a number of occurrences of the possible outputvalue in the layer output values (220). In exemplary implementations,when the predetermined set of possible output values includes integervalues in the range [0, 7], the system counts the number of occurrenceseach of those integer values in the layer output values. For instance,if the integer 0 has appeared five times in the layer output values, thesystem determines that count(0)=5.

The system determines an exponentiation measure for each possible outputvalue occurring at least once in the layer output values (230). That is,for each x that is one of the possible output values in thepredetermined finite set, if the count of x among the layer outputvalues is one or more, then the system determines an exponentiationmeasure of x.

Generally, the system determines the respective exponentiation measureof a given possible output value by exponentiating Euler's number (i.e.the number e) by the possible output value or by the result of themultiplication of the possible output value by another value. In someimplementations, the system determines the respective exponentiationmeasure of each respective possible output value by computing e^X, whereX is the respective possible output value. In some otherimplementations, the system determines the respective exponentiationmeasure of each respective possible output value by computing e^(X/T),where X is the respective possible output value and T is a temperaturevalue.

In some implementations, the system performs the exponentiationoperation on the output of a function, such as an A-law mappingfunction, a μ-law mapping function, or a compression function.

In some implementations, the system has pre-computed a respectiveexponentiation measure for each of the possible output values andaccesses the corresponding precomputed value to determine theexponentiation measure of the possible output value.

The system determines a normalization factor (i.e., the denominator ofthe softmax function) for the layer output by combining theexponentiation measures in accordance with the number of occurrences ofthe possible output values (240). In some implementations, for eachmember of the set of the possible output values in the predeterminedfinite set whose corresponding count is one or more, the systemmultiplies the count for the member by the exponentiation measure forthe member. The system then generates the normalization factor by addingthe results of those multiplications.

The system determines a softmax probability value for each of the layeroutput values from the respective exponentiation measure for the layeroutput value and the normalization factor (250). In particular, thesystem determines each softmax probability value for a given layeroutput value by dividing the exponentiation measure for the layer outputvalue by the normalization factor.

In some implementations, each softmax probability value in the networkoutput represents a normalized likelihood of a corresponding conditionbeing satisfied.

For example, when the neural network generates scores for objectcategories, each softmax probability value corresponds to a differentobject category and represents the normalized likelihood that the imageincludes an object belonging to the category.

As another example, when the neural network generates scores for topicsof Internet resources, documents, or portions of documents, each softmaxprobability value corresponds to a different topic and represents thenormalized likelihood that the Internet resource, document, or portionof document is about the topic.

As another example, when the neural network generates scoresrepresenting the estimated likelihood that advertisements will beclicked on, each softmax probability value represents the normalizedlikelihood that a particular advertisement will be clicked on.

FIG. 3 is a flow chart of an example 300 for mapping initial layeroutput values to layer output values. For convenience, the process 300will be described as being performed by a system of one or morecomputers located in one or more locations. For example, a neuralnetwork processing system, e.g., the neural network processing system100 of FIG. 1, appropriately programmed in accordance with thisspecification, can perform the process 300.

The system receives initial layer output values from a processing systemthat does the processing for the one or more initial neural networklayers (310). The initial layer output values are unmapped output valuesof the one or more initial neural network layers.

The system obtains mapping data defining a mapping from the initiallayer output values to the layer output values (320). In exemplaryimplementations, when the predetermined set of possible output valuesincludes integer values in the range [0, 7], the system obtains datamapping 0 to π, 1 to 2π, 2 to 3π, and so on (i.e. for each integer valuex, the system maps x to (x+1)*π). In some implementations, the mappingdata specifies a scaling factor for scaling each of the initial layeroutput values to generate the layer output values. A scaling factor is anumber by which an initial layer output value will be multiplied togenerate a corresponding layer output value.

The system determines a layer output value for each initial layer outputvalue based on the mapping data (330). If the mapping data specify alayer output value corresponding to an initial layer output value, thesystem uses that specified value to determine the layer output value. Ifthe mapping data specify a scaling factor by which a layer output valueis determined based on the initial layer output value, the system scalesthe initial layer output value with the scaling factor to determine thelayer output value.

Embodiments of the subject matter and the functional operationsdescribed in this specification can be implemented in digital electroniccircuitry, in tangibly-embodied computer software or firmware, incomputer hardware, including the structures disclosed in thisspecification and their structural equivalents, or in combinations ofone or more of them. Embodiments of the subject matter described in thisspecification can be implemented as one or more computer programs, i.e.,one or more modules of computer program instructions encoded on atangible non-transitory storage medium for execution by, or to controlthe operation of, data processing apparatus. The computer storage mediumcan be a machine-readable storage device, a machine-readable storagesubstrate, a random or serial access memory device, or a combination ofone or more of them. Alternatively, or in addition, the programinstructions can be encoded on an artificially-generated propagatedsignal, e.g., a machine-generated electrical, optical, orelectromagnetic signal, that is generated to encode information fortransmission to suitable receiver apparatus for execution by a dataprocessing apparatus.

The term “data processing apparatus” refers to data processing hardwareand encompasses all kinds of apparatus, devices, and machines forprocessing data, including by way of example a programmable processor, acomputer, or multiple processors or computers. The apparatus can alsobe, or further include, special purpose logic circuitry, e.g., an FPGA(field programmable gate array) or an ASIC (application-specificintegrated circuit). The apparatus can optionally include, in additionto hardware, code that creates an execution environment for computerprograms, e.g., code that constitutes processor firmware, a protocolstack, a database management system, an operating system, or acombination of one or more of them.

A computer program which may also be referred to or described as aprogram, software, a software application, an app, a module, a softwaremodule, a script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, or declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a computing environment. A program may, but neednot, correspond to a file in a file system. A program can be stored in aportion of a file that holds other programs or data, e.g., one or morescripts stored in a markup language document, in a single file dedicatedto the program in question, or in multiple coordinated files, e.g.,files that store one or more modules, sub-programs, or portions of code.A computer program can be deployed to be executed on one computer or onmultiple computers that are located at one site or distributed acrossmultiple sites and interconnected by a data communication network.

For a system of one or more computers to be configured to performparticular operations or actions means that the system has installed onit software, firmware, hardware, or a combination of them that inoperation cause the system to perform the operations or actions. For oneor more computer programs to be configured to perform particularoperations or actions means that the one or more programs includeinstructions that, when executed by data processing apparatus, cause theapparatus to perform the operations or actions.

As used in this specification, an “engine,” or “software engine,” refersto a software implemented input/output system that provides an outputthat is different from the input. An engine can be an encoded block offunctionality, such as a library, a platform, a software development kit(“SDK”), or an object. Each engine can be implemented on any appropriatetype of computing device, e.g., servers, mobile phones, tabletcomputers, notebook computers, music players, e-book readers, laptop ordesktop computers, PDAs, smart phones, or other stationary or portabledevices, that includes one or more processors and computer readablemedia. Additionally, two or more of the engines may be implemented onthe same computing device, or on different computing devices.

The processes and logic flows described in this specification can beperformed by one or more programmable computers executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby special purpose logic circuitry, e.g., an FPGA or an ASIC, or by acombination of special purpose logic circuitry and one or moreprogrammed computers.

Computers suitable for the execution of a computer program can be basedon general or special purpose microprocessors or both, or any other kindof central processing unit. Generally, a central processing unit willreceive instructions and data from a read-only memory or a random accessmemory or both. The essential elements of a computer are a centralprocessing unit for performing or executing instructions and one or morememory devices for storing instructions and data. The central processingunit and the memory can be supplemented by, or incorporated in, specialpurpose logic circuitry. Generally, a computer will also include, or beoperatively coupled to receive data from or transfer data to, or both,one or more mass storage devices for storing data, e.g., magnetic,magneto-optical disks, or optical disks. However, a computer need nothave such devices. Moreover, a computer can be embedded in anotherdevice, e.g., a mobile telephone, a personal digital assistant (PDA), amobile audio or video player, a game console, a Global PositioningSystem (GPS) receiver, or a portable storage device, e.g., a universalserial bus (USB) flash drive, to name just a few.

Computer-readable media suitable for storing computer programinstructions and data include all forms of non-volatile memory, mediaand memory devices, including by way of example semiconductor memorydevices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks,e.g., internal hard disks or removable disks; magneto-optical disks; andCD-ROM and DVD-ROM disks.

To provide for interaction with a user, embodiments of the subjectmatter described in this specification can be implemented on a computerhaving a display device, e.g., a CRT (cathode ray tube) or LCD (liquidcrystal display) monitor, for displaying information to the user and akeyboard and pointing device, e.g., a mouse, trackball, or a presencesensitive display or other surface by which the user can provide inputto the computer. Other kinds of devices can be used to provide forinteraction with a user as well; for example, feedback provided to theuser can be any form of sensory feedback, e.g., visual feedback,auditory feedback, or tactile feedback; and input from the user can bereceived in any form, including acoustic, speech, or tactile input. Inaddition, a computer can interact with a user by sending documents toand receiving documents from a device that is used by the user; forexample, by sending web pages to a web browser on a user's device inresponse to requests received from the web browser. Also, a computer caninteract with a user by sending text messages or other forms of messageto a personal device, e.g., a smartphone, running a messagingapplication, and receiving responsive messages from the user in return.

Embodiments of the subject matter described in this specification can beimplemented in a computing system that includes a back-end component,e.g., as a data server, or that includes a middleware component, e.g.,an application server, or that includes a front-end component, e.g., aclient computer having a graphical user interface, a web browser, or anapp through which a user can interact with an implementation of thesubject matter described in this specification, or any combination ofone or more such back-end, middleware, or front-end components. Thecomponents of the system can be interconnected by any form or medium ofdigital data communication, e.g., a communication network. Examples ofcommunication networks include a local area network (LAN) and a widearea network (WAN), e.g., the Internet.

The computing system can include clients and servers. A client andserver are generally remote from each other and typically interactthrough a communication network. The relationship of client and serverarises by virtue of computer programs running on the respectivecomputers and having a client-server relationship to each other. In someembodiments, a server transmits data, e.g., an HTML page, to a userdevice, e.g., for purposes of displaying data to and receiving userinput from a user interacting with the device, which acts as a client.Data generated at the user device, e.g., a result of the userinteraction, can be received at the server from the device.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of anyinvention or on the scope of what may be claimed, but rather asdescriptions of features that may be specific to particular embodimentsof particular inventions. Certain features that are described in thisspecification in the context of separate embodiments can also beimplemented in combination in a single embodiment. Conversely, variousfeatures that are described in the context of a single embodiment canalso be implemented in multiple embodiments separately or in anysuitable subcombination. Moreover, although features may be describedabove as acting in certain combinations and even initially be claimed assuch, one or more features from a claimed combination can in some casesbe excised from the combination, and the claimed combination may bedirected to a subcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various system modulesand components in the embodiments described above should not beunderstood as requiring such separation in all embodiments, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Particular embodiments of the subject matter have been described. Otherembodiments are within the scope of the following claims. For example,the actions recited in the claims can be performed in a different orderand still achieve desirable results. As one example, the processesdepicted in the accompanying figures do not necessarily require theparticular order shown, or sequential order, to achieve desirableresults. In certain some cases, multitasking and parallel processing maybe advantageous.

What is claimed is:
 1. A method of processing a network input through aneural network having one or more initial neural network layers followedby a softmax output layer to generate a neural network output for thenetwork input, wherein the initial neural network layers are implementedby a processing system that performs computations specified by the oneor more initial neural network layers using quantized arithmetic suchthat output values generated by the processing system can take onlyvalues from a predetermined finite set of values, the method comprising:precomputing a respective exponentiation measure for each possibleoutput value of a predetermined finite set of possible output valuesthat can be included in layer outputs generated by processing networkinputs through the one or more initial neural network layers using theprocessing system; storing the respective precomputed exponentiationmeasures; obtaining a first layer output generated by the processingsystem by processing a first network input through the one or moreinitial neural network layers, the first layer output having a pluralityof first layer output values, and each first layer output value being arespective one of the predetermined finite set of possible outputvalues; and processing the layer output through the softmax output layerto generate a first neural network output for the network input,comprising: determining, for each possible output value in thepredetermined finite set of possible output values, a count ofoccurrences of the possible output value among the plurality of firstlayer output values that are in the first layer output; for eachpossible output value that occurs at least once in the in the pluralityof layer output values, accessing the stored respective precomputedexponentiation measure for the possible output value, instead ofre-computing the respective exponentiation measure; determining anormalization factor for the layer output by combining the storedprecomputed exponentiation measures for the possible output values thatoccur in the first layer output at least once in accordance with thecounts of occurrences of the possible output values; and determining,for each of the plurality of first layer output values, a softmaxprobability value from the respective precomputed exponentiation measurefor the first layer output value and the normalization factor.
 2. Themethod of claim 1, wherein obtaining the first layer output comprises:receiving a plurality of initial layer output values from the processingsystem, the plurality of initial layer output values being unmappedoutput values of the one or more initial neural network layers that areeach a respective one of the predetermined finite set of values;obtaining mapping data defining a mapping from the predetermined finiteset of values to the predetermined finite set of possible layer outputvalues; and determining, for each initial layer output value, acorresponding first layer output value based on the mapping data.
 3. Themethod of claim 2, wherein the mapping data specifies a scaling factorfor scaling each of the plurality of initial layer output values togenerate the layer output values.
 4. The method of claim 1, wherein eachof the finite set of possible output values map to a respective value ofan integer data type.
 5. The method of claim 1, further comprising:generating the first network input by converting one or more floatingpoint values to fixed point values.
 6. The method of claim 1, whereindetermining the respective precomputed exponentiation measure for eachpossible output value comprises exponentiating Euler's number by amultiplication of the possible output value.
 7. The method of claim 1,wherein each softmax probability value is determined by dividing eachrespective precomputed exponentiation measure by the normalizationfactor.
 8. The method of claim 1, wherein each of the finite set ofpossible output values is an output of a mapping function.
 9. The methodof claim 1, wherein each of the finite set of possible output values isan output of a compression function.
 10. A system comprising one or morecomputers and one or more storage devices storing instructions that areoperable, when executed by the one or more computers, to cause the oneor more computers to perform operations for processing a network inputthrough a neural network having one or more initial neural networklayers followed by a softmax output layer to generate a neural networkoutput for the network input, wherein the initial neural network layersare implemented by a processing system that performs computationsspecified by the one or more initial neural network layers usingquantized arithmetic such that output values generated by the processingsystem can take only values from a predetermined finite set of values,the operations comprising: precomputing a respective exponentiationmeasure for each possible output value of a predetermined finite set ofpossible output values that can be included in layer outputs generatedby processing network inputs through the one or more initial neuralnetwork layers using the processing system; storing the respectiveprecomputed exponentiation measures; obtaining a first layer outputgenerated by the processing system processing a first network inputthrough the one or more initial neural network layers, the first layeroutput having a plurality of first layer output values, and each firstlayer output value being a respective one of the predetermined finiteset of possible output values; and processing the layer output throughthe softmax output layer to generate a first neural network output forthe network input, comprising: determining, for each possible outputvalue in the predetermined finite set of possible output values, a countof occurrences of the possible output value among the plurality of firstlayer output values that are in the first layer output; for eachpossible output value that occurs at least once in the in the pluralityof layer output values, accessing the stored precomputed exponentiationmeasure for the possible output value, instead of re-computing therespective exponentiation measure; determining a normalization factorfor the layer output by combining the stored precomputed exponentiationmeasures for the possible output values that occur in the first layeroutput at least once in accordance with the counts of occurrences of thepossible output values; and determining, for each of the plurality offirst layer output values, a softmax probability value from therespective precomputed exponentiation measure for the first layer outputvalue and the normalization factor.
 11. The system of claim 10, whereinobtaining the first layer output comprises: receiving a plurality ofinitial layer output values from the processing system, the plurality ofinitial layer output values being unmapped output values of the one ormore initial neural network layers that are each a respective one of thepredetermined finite set of values; obtaining mapping data defining amapping from the predetermined finite set of values to the predeterminedfinite set of possible layer output values; and determining, for eachinitial layer output value, a corresponding first layer output valuebased on the mapping data.
 12. The system of claim 11, wherein themapping data specifies a scaling factor for scaling each of theplurality of initial layer output values to generate the layer outputvalues.
 13. The system of claim 10, wherein each of the finite set ofpossible output values map to a respective value of an integer datatype.
 14. The system of claim 10, further comprising: generating thefirst network input by converting one or more floating point values tofixed point values.
 15. The system of claim 10, wherein determining therespective precomputed exponentiation measure for each possible outputvalue comprises exponentiating Euler's number by a multiplication of thepossible output value.
 16. A non-transitory computer storage mediumencoded with instructions that, when executed by one or more computers,cause the one or more computers to perform operations for processing anetwork input through a neural network having one or more initial neuralnetwork layers followed by a softmax output layer to generate a neuralnetwork output for the network input, wherein the initial neural networklayers are implemented by a processing system that performs computationsspecified by the one or more initial neural network layers usingquantized arithmetic such that output values generated by the processingsystem can take only values from a predetermined finite set of values,the operations comprising: precomputing a respective exponentiationmeasure for each possible output value of a predetermined finite set ofpossible output values that can be included in layer outputs generatedby processing network inputs through the one or more initial neuralnetwork layers using the processing system; storing the respectiveprecomputed exponentiation measures; obtaining a first layer outputgenerated by the processing system processing a first network inputthrough the one or more initial neural network layers, the first layeroutput having a plurality of first layer output values, and each firstlayer output value being a respective one of the predetermined finiteset of possible output values; and processing the layer output throughthe softmax output layer to generate a first neural network output forthe network input, comprising: determining, for each possible outputvalue in the predetermined finite set of possible output values, a countof occurrences of the possible output value among the plurality of firstlayer output values that are in the first layer output; for eachpossible output value that occurs at least once in the in the pluralityof layer output values, accessing the stored precomputed exponentiationmeasure for the possible output value, instead of re-computing therespective exponentiation measure; determining a normalization factorfor the layer output by combining the stored precomputed exponentiationmeasures for the possible output values that occur in the first layeroutput at least once in accordance with the counts of occurrences of thepossible output values; and determining, for each of the plurality offirst layer output values, a softmax probability value from therespective precomputed exponentiation measure for the first layer outputvalue and the normalization factor.
 17. The non-transitory computerstorage medium of claim 16, wherein obtaining the first layer outputcomprises: receiving a plurality of initial layer output values from theprocessing system, the plurality of initial layer output values beingunmapped output values of the one or more initial neural network layersthat are each a respective one of the predetermined finite set ofvalues; obtaining mapping data defining a mapping from the predeterminedfinite set of values to the predetermined finite set of possible layeroutput values; and determining, for each initial layer output value, acorresponding first layer output value based on the mapping data. 18.The non-transitory computer storage medium of claim 17, wherein themapping data specifies a scaling factor for scaling each of theplurality of initial layer output values to generate the layer outputvalues.
 19. The non-transitory computer storage medium of claim 16,wherein each of the finite set of possible output values map to arespective value of an integer data type.
 20. The non-transitorycomputer storage medium of claim 16, wherein determining the respectiveprecomputed exponentiation measure for each possible output valuecomprises exponentiating Euler's number by a multiplication of thepossible output value.